To determine which of the given equations represent [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex], we need to see if we can express [tex]\( y \)[/tex] uniquely in terms of [tex]\( x \)[/tex] for each equation.
1. Equation (1): [tex]\( x y = -8 \)[/tex]
We can solve this for [tex]\( y \)[/tex]:
[tex]\[
xy = -8 \implies y = \frac{-8}{x}
\][/tex]
Here, [tex]\( y \)[/tex] is explicitly expressed as a function of [tex]\( x \)[/tex]. Therefore, equation (1) represents [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex].
2. Equation (2): [tex]\( 4 x^2 + 9 y^2 = 36 \)[/tex]
We can attempt to solve this for [tex]\( y \)[/tex]:
[tex]\[
4 x^2 + 9 y^2 = 36 \implies 9 y^2 = 36 - 4 x^2 \implies y^2 = \frac{36 - 4 x^2}{9} \implies y = \pm \frac{\sqrt{36 - 4 x^2}}{3}
\][/tex]
This yields two solutions for [tex]\( y \)[/tex] (one positive and one negative) for each [tex]\( x \)[/tex] in the respective domain. Consequently, [tex]\( y \)[/tex] is not uniquely determined by [tex]\( x \)[/tex], so equation (2) does not represent [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex].
3. Equation (3): [tex]\( 3 x^2 - y = 1 \)[/tex]
Solving for [tex]\( y \)[/tex]:
[tex]\[
3 x^2 - y = 1 \implies y = 3 x^2 - 1
\][/tex]
Here, [tex]\( y \)[/tex] is explicitly expressed as a function of [tex]\( x \)[/tex]. Therefore, equation (3) represents [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex].
4. Equation (4): [tex]\( y^2 - x^2 = 4 \)[/tex]
We can attempt to solve this for [tex]\( y \)[/tex]:
[tex]\[
y^2 - x^2 = 4 \implies y^2 = x^2 + 4 \implies y = \pm \sqrt{x^2 + 4}
\][/tex]
This yields two solutions for [tex]\( y \)[/tex] (one positive and one negative) for each [tex]\( x \)[/tex]. Consequently, [tex]\( y \)[/tex] is not uniquely determined by [tex]\( x \)[/tex], so equation (4) does not represent [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex].
In summary, the equations that represent [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex] are:
- Equation (1)
- Equation (3)
